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I'm working on this problem and I'm having difficulty understanding the solution. In the solution, it states $a^3+b^3=(a+b)^3\rightarrow ab(a+b)=0$. What do these equations mean? It doesn't seem to fit the identity $a^3+b^3=(a+b)(a^2-ab+b^2)$ if the arrow meant subtracting.

I know that right arrows can define a function, but I don't see how $(a+b)^3\rightarrow ab(a+b)=0$ could be a function. If it is, then I don't know how it is applicable to the problem, in which case it would be great if someone could explain the rest of the solution.

P. S. I don't see this as asking two questions if that's not allowed. One is just a prerequisite to the other.

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  • $\begingroup$ The argument is asserting that $a^3+ b^3 = ( a+b)^3$, then indicating that this relation leads to ("$\to$") $ab(a+b)=0$. That kind of notation probably makes more sense in displayed equations like $$a^3+ b^3 = ( a+b)^3 \qquad\to\qquad ab(a+b)=0$$ (At least, I hope it makes sense ... I do this all the time!) Incorporating it into text probably isn't the best presentation. $\endgroup$ – Blue Nov 9 at 5:20
  • $\begingroup$ $“A\to B”$ means that whenever $A$ is true, then $B$ is also true. $\endgroup$ – MJD Nov 9 at 15:11
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Assume that $$ a^3 + b^3 = \left( {a + b} \right)^3 $$ Since $$ \left( {a + b} \right)^3 = a^3 + b^3 + 3a^2 b + 3ab^2 $$ it follows that $$ 3a^2 b + 3ab^2 = 0 $$ thus $$ 3ab\left( {a + b} \right) = 0 $$ so that $$ ab\left( {a + b} \right) = 0 $$

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It means: if $a^3+b^3 =(a+b)^3$, then $ab (a+b)=0$.

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In this context, the arrow is a symbol which describes the logical flow of the argument being made. It should be read as "implies" or "leads to". That is, if $A$ and $B$ are two statements, then the notation $$A \to B$$ can be read "$A$ implies $B$", or "$A$ leads to $B$". Alternatively, this can also be read as "If $A$ is a true statement, then $B$ is a true statement as well", or, more succinctly, "If $A$, then $B$".

In the context of the particular argument presented in the question, the notation $$ a^3+b^3=(a+b)^3\rightarrow ab(a+b)=0 $$ means that the truth of the first equation implies the truth of the second equation. Written with sightly less notation, we might say

If $a^3 + b^3 = (a+b)^3$, then $ab(a+b)=0$.

This answer shows how one might obtain that result. Written more compactly, one might write

\begin{align} &a^3 + b^3 = (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 && \text{(expand the RHS)} \\ &\qquad\rightarrow 0 = 3a^2b + 3ab^2 = 3ab(a^2+b^2) && \text{(cancel $a^3+b^3$, factor)} \\ &\qquad\rightarrow 0 = ab(a^2 + b^2). && \text{(cancel $3$)} \end{align}

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the arrow means it follows: just calculate (a+b)^3 and subtract a^3+b^3 what remains is 0

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  • $\begingroup$ This does not provide an answer to the question. To critique or request clarification from an author, leave a comment below their post. - From Review $\endgroup$ – Klangen Nov 9 at 5:45
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    $\begingroup$ @Klangen, I think it does answer the question. The arrow means from the first it follows the second. One must expand the LHS and substract the RHS from it. $\endgroup$ – farruhota Nov 9 at 7:51

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